Near-crack contour behaviour and extraction of log-singular stress terms of the self-regular traction boundary integral equation

This work contains an analytical study of the asymptotic near-crack contour behaviour of stresses obtained from the self-regular traction-boundary integral equation (BIE), both in two and in three dimensions, and for various crack displacement modes. The at crack case is chosen for detailed analysis of the singular stress for points approaching the crack contour. By imposing a condition of bounded stresses on the crack surface, the work shows that the boundary stresses on the crack are in fact zero for an unloaded crack, and the interior stresses reproduce the known inverse square root behaviour when the distance from the interior point to the crack contour approaches zero. The correct order of the stress singularity is obtained after the integrals for the self-regular traction-BIE formulation are evaluated analytically for the assumed displacement discontinuity model. Based on the analytic results, a new near-crack contour self-regular traction-BIE is proposed for collocation points near the crack contour. In this new formulation, the asymptotic log-singular stresses are identi ed and extracted from the BIE. Log-singular stress terms are revealed for the free integrals written as contour integrals and for the self-regularized integral with the integration region divided into sub-regions. These terms are shown to cancel each other exactly when combined and can therefore be eliminated from the nal BIE formulation. This work separates mathematical and physical singularities in a unique manner. Mathematical singularities are identi ed, and the singular information is all contained in the region near the crack contour.